THE ELECTRONIC JOURNAL OF COMBINATORICS (2014), DS1.14 References

and Computing 11. The results of 143 references depend on computer algorithms. The references are ordered alphabetically by the last name of the first author, and where multiple papers have the same first author they are ordered by the last name of the second author, etc. We preferred that all work by the same author be in consecutive positions. Unfortunately, this causes that some of the abbreviations are not in alphabetical order. For example, [BaRT] is earlier on the list than [BaLS]. We also wish to explain a possible confusion with respect to the order of parts and spelling of Chinese names. We put them without any abbreviations, often with the last name written first as is customary in original. Sometimes this is different from the citations in other sources. One can obtain all variations of writing any specific name by consulting the authors database of Mathematical Reviews a

Total Graph Interpretation of the Numbers of the Fibonacci Type

We give a total graph interpretation of the numbers of the Fibonacci type. This graph interpretation relates to an edge colouring by monochromatic paths in graphs. We will show that it works for almost all numbers of the Fibonacci type. Moreover, we give the lower bound and the upper bound for the number of all (A1,2A1)-edge colourings in trees

Una Lampada nella notte [:] L'Ars inventiva per triginta statuas di Giordano Bruno

The subject of this thesis, i.e. Giordano Bruno's treatise, known as 'Lampas triginta statuarum', is a text that remained inedited for a long time and was published only in 1891 by Felice Tocco and Girolamo Vitelli in Opera latine conscripta third volume. This philosophic work was written by Bruno during his stay in Wittemberg (1586-88) when he, working as a teacher in Wittemberg Academy, lectured on Aristotle's Organon. During those years he wrote and sent to press other two 'Lampadi' behind this: De Lampade combinatoria Lulliana (a Lullo's Ars Magna comment) and De progressu et Lampade venatoria logicorum (Topici compendium). With Lampas triginta statuarum, that completes the Thrilogy, he intended to improve traditional logical instruments, those derived from Aristotelian and "Porphirian" tradition as well as those derived from Lullian tradition. This treatise was conveyed in its first redaction by Augsburn Codex (A), together with printed test of De Lampade combinatoria Lulliana and with the manuscript Animadversiones circa Lampadem Lullianam, written using the same paper and the same ink already used for the Lampas by a unknown copyist, who was a German Bruno's student. Codex A, preserved in Augsburn Bibliothec, probably belongs to Heinrich Heinzel to whom Bruno dedicated last published work, De imaginum, signorum, et idearum compositione (Francoforte, 1591), that, although written later, can be for many aspects related to Lampas. Having left Wittemberg in 1588, and living in Helmsted (1589 - 90), Bruno devoted himself to the composition of some texts (De magia e Theses de magia, De rerum principiis et elementis et causis, Medicina Lulliana, De magia Mathematica), that remained unedited and were collected in Moscow Codex (M) together with a second manuscript of Lampas triginta statuarum and with De Vinculis in genere. Moscow Codex, also named Norov Codex from the Russian nobleman Avraam Sergeevic Norov that acquired it for his collection from the bookseller Edwin Tross, contains some autograph pages, but it's mostly written by Hyeronimus Besler. In the autumn 1591, while Bruno was living in Padua, Besler transcribed De Vinculis, drafted in Frankfurt, and copied the treatise 'ars inventiva per triginta statuas'. In this work the philosopher leaves the harsh tone used to polemize with his contemporary followers of Aristotle's philosophy, to open to a dialog and a tight mach with the tradition, in particular with the Aristotelian-scholastic one, through a deep analysis of thirty concepts used in philosophic practice, represented by thirty powerful mythologic figures. From this point of view, the treatise is structured like a dictionary of philosophic terms (Bruno appears to be inspired by Aristotle's Metafisica V book.) and also shows some affinity with another text, that was composed some time later in Zurich and that remained unedited (this text was published as a posthumus work by Bruno's student Raphael Egli in two following editions, the first one in 1595 and the second in 1609 comprehensive of the section Praxis descensus seu applicatio entis): it's the Summa terminorum metaphysicorum, a "philosophic nomenclature treatise" as defined by Erminio Troilo) in which fifty two conceptual categories belonging to Aristotelian scholastic lexicom are deeply analysed. In Centum et viginti articuli de natura et mundo adversus Peripateticos (the programmatic phamplet dealing with Cambrai disputation that took place in Paris in 1586), also taken up in Acrotismus (edited in Praga in 1588), the criticism of the natural Aristotle's philosophy is developed through theses regarding some fundamental concepts such as nature, movement, time, place...and so on. In Lampas Bruno goes on in this re-definition of philosophic terms, already started in De la causa, principio et uno: he translates his revolutionary philosophic conceptions into tradition lexicon and in doing this he moulds the meanings and transforms them, joining them in a new plot. In "thirdy statues treatise" the philosopher seems to be moved by a strong systematic intent: that is by the need to proceed in a rigorous and scientific way in the exposition of the foundation of the 'nova filosofia'. Knowledge architecture, within which speculative contents are placed is an imaginary museum filled with statues/concepts: Apollo (unitas), Saturn (principium), Prometheus (agens), Vulcan (forma), Thetis (causa materialis), Sagittarius (causa finalis) and Mount Olympus (finis)...and so on. In moulding the statues, taking from the wide mithology catalogue Bruno uses the "Fidia's seal" (Explicatio triginta sigillorum), as Frances Amelia Yates first guessed (The Art of Memory, 1966). In this thesis my intent is to focus on Bruno's ars memoriae original characters as well as on the forms in which "seals art" is used in Lampas, developing F.A. Yates's insight. One of the most fundamental figure appears to be Prometheus, the imagination architect, the sculptor that first moulded clay statues breathing into them the spiritus. In Bruno's texts, specifically in Lampas and De Imaginum Prometheus became the archetype of the imagination activity, the symbol of the memory artist's praxis. A special emphasis is given to the analysis of the ways used by Bruno to compose the images (consisting in very detailed verbal descriptions) through the research of both literary and iconologic sources (from Boccaccio to Ovidio to the Renaissance mythographers Vincenzo Cartari e Natale Conti). Bruno's work is deeply permeated and inspirited by a didactic intent that is here mostly considered: i.e. the figuration of the most abstract concepts by means of sensitive statues, that appear to sight and mind exploration. The treatise 'ars inventiva per triginta statuas' intends to be mainly a new modus docendi a way of "thinking through images" that is in the same time a way of "thinking through concepts". The "discursiva architectura", hosting the statues must not be considered static or motionless, like physical buildings: in fact the statues can be moved and joined together in many different ways, depending on everybody's own ingenium. Bruno's images are imagini agentes, in other words they have a special effectiveness from the emotional point of view, having the ability to activate cognitive processes. Bruno's opera distinctive features, Lamp structure and functioning, characters and purposes of Bruno's complex logical linguistic system are gradually outlined in this thesis, through text analysis and sources examination. The first chapter presents the great framework inclosing statues gallery: two infigurabilia triads that together represent Bruno's architecture fundamentals and metaphysical basis for ars inventiva that is here applied. In the second chapter "statues and trees" you are introduced to Bruno's use of "statua" mnemonic artifice. By means of the first sequence of statues (Apollo, Saturn, Prometheus, Vulcan, Thetis, Sagittarius and Mount Olympus), it's explained the whole process in natural and human creation. The lexical reference plot is reviewed, with specific consideration to Aristotle's Metafisica fifth Book. "Tree" image in Bruno's texts and "tree" use in the section dedicated to the logic in Lampas treatise is examined for what concerns its relations with both Lullo's and Aristotelian scholastic tradition. Creative activity is represented by Prometheus, the sculptor of alive statues, the creator of natural and artificial things. In the third chapter "Prometheus myth in Lampas triginta statuarum", a deep analysis is carried on about the Statue dedicate by Bruno to Titan, with the purpose to highlight its special relevance in the complete framework. In the two statues dedicated to Minerva (De campo Minervae, seu de noticia e De schala Minervae, seu de habitibus cognitionis) Bruno doesn't provide any goddess image. To Minerva's portrait, outlined by Bruno in Oratio Valedictoria is dedicated the last section (Minerva in light shadow). In Lampas treatise Bruno puts into practice his art of memory, deeply rooted in the living art of nature and represents a new "methodus", suitable for human mind's cognitive structures. The relation among art of memory, theory of knowledge and doctrine of soul is a complex problem of which I pointed out the most important aspects and that in my opinion and in my wishes would be worth further analysing. Beside the figure of the ingenious architect and of the cultivator of imagination "campus", it comes out in my work the image of the other Bruno, the "philosopher", the systematic and rigorous thinker and the fine connoisseur of Aristotelian doctrines, although by himself severely criticized

Una fonte trascurata sul rapporto tra qabbalah e combinatoria lulliana in Pico della Mirandola: il commento alle preghiere di Yehudah Ibn Malka*

L’articolo propone una discussione su nuove basi documentarie sui rapporti, spesso evocati, tra ars brevis lulliana e speculazioni cabbalistiche. L’associazione anaogica tra Lullo e la qabbalah è stata proposta da Giovanni Pico della Mirandola nella sua Apologia del 1487 ma in termini, come suo consueto, piuttosto vaghi. Quel che, sino a pochi anni fa, non era noto è che Pico dispoeva di un ampio frammento, nella traduzione latina di Flavio Mitridate, del Commento alle preghiere di Yehudah ben Nissim Ibn Malka, coevo del filosofo francescano, in cui si propne un metodo esegetico cabbaistico imperniato sull’uso di ruote alfabetiche concentriche. Si esamina la possibilità che l’origine dell’analogia e della distinzione proposta da Pico sia da ricercare in questo testo, sfiorando anche la questione della possibilità reale di un influsso storico alle sorgenti del lullismo. In appendice si pubblica anche l’edizione del testo latino che fu letto e glossato da Pico, per favorirne la riemersione nel dibattito critico dopo secoli di oblioOn the basis of new documentation, this article proposes a discussion of the often cited relations between the Lullian Ars brevis and Kabbalistic speculation. The analogic association between Llull and the Kabbalah was proposed by Giovanni Pico della Mirandola in his Apologia of 1487, but in rather vague terms, as was usual with him. What was not known until a few years ago was that Pico has access to an extended fragment of the Latin translation by Flavius Mithridates of the Commentary on the prayers of Yehudah ben Nissim Ibn Malka, a contemporary of the Franciscan philosopher, in which is proposed an exegetical Kabbalistic method based on the use of concentric alphabetical circles. The article explores the possibility that the origin of the analogy and distinction proposed by Pico might be found in this text, as well as touching on the question of the real possibility of a historical influence on the sources of Lullism. In an Appendix is published an edition of the Latin text which was read and glossed by Pico, in order to contribute to the reemergence of the critical debate after centuries of neglect